Stephen P. King (email@example.com)
Thu, 02 Sep 1999 22:56:09 -0400
My friend Paul Hanna pointed me to this page. It is very similar to
what he is working on!
A three-dimensional space that is everywhere locally Eulcidean and yet cylindrical in all directions can be constructed by embedding the three spatial dimensions in a six-dimensional space according to the parameterization x1 = R1 cos x/R1 x2 = R1 sin x/R1 x3 = R2 cos y/R2 x4 = R2 sin y/R2 x5 = R3 cos z/R3 x6 = R3 sin z/R3 so the spatial Euclidean line element is dx1^2 + dx2^2 + dx3^2 + dx4^2 + dx5^2 + dx6^2 = dx^2 + dy^2 + dz^2 giving a Euclidean spatial metric in a closed three-space with total volume (2*pi)^3*R1*R2*R3. It's been suggested (by Klaus Kassner, among others) that we can subtract this from an ordinary temporal component to give an everywhere-locally-Lorentzian spacetime that is cylindrical in the three spatial directions, i.e., ds^2 = c^2 dt^2 - (dx^2 + dy^2 + dz^2) However, this "subtraction" doesn't seem well motivated. One way of providing a motivation, and in the process making the universe cylindrical in ALL directions, temporal as well as spatial, would be to embed the entire 4D spacetime into a space of 8 dimensions, 2 of which are purely imaginary, like this x1 = R1 cos x/R1 x2 = R1 sin x/R1 x3 = R2 cos y/R2 x4 = R2 sin y/R2 x5 = R3 cos z/R3 x6 = R3 sin z/R3 x7 = i R4 cos t/R4 x8 = i R4 sin t/R4 leading (again) to a locally Lorentzian 4D metric (ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2 but now *all four* of the dimensions x,y,z,t are periodic. So here we have an everywhere-locally-Lorentzian manifold that is closed and unbounded in every spatial and temporal direction. (Obviously this manifold contains closed time-like worldlines.) This reminds me a bit of Stephen Hawking's recent attempts to describe a universe that is finite but unbounded in time as well as space. He too invokes an imaginary time dimension, but aside from that I don't think his model is related to the locally-flat cosmology described here. Anyway, the causal structure of such a universe is interesting. We might imagine that a flat, closed, unbounded universe of this type would tend to collapse if it contained any matter, unless a non-zero cosmological constant is assumed. On the other hand, I'm not sure what "collapse" would mean in this context. It might mean that the R parameters would shrink, but R is not a dynamical parameter of the model. The 4D field equations operate only on x,y,z,t. Also, any "change" in R would imply some meta-time parameter T, so that all the R coefficients in the embedding formulas would actually be functions R(T). It seems that the flatness of the 4-space is independent of the value of R(T), and if the field equations are satisfied for one value of R they would be satisfied for any value of R. But I'm not sure how the meta-time T would relate to the internal time t for a given observer. It might require some "meta field equations" to relate T to the internal parameters x,y,z,t. Possibly these meta-equations would allow (require?) the value of R to be "increasing" versus T, and therefore indirectly versus our internal time t = f(T), in order to achieve stability. On the general method of embedding a locally flat n-dimensional space in a flat 2n-dimensional space, I wonder if every orthogonal basis in the n-space maps to an orthogonal basis in the 2n-space according to a set of formulas formally the same as those shown above. If not, is there a more general mapping that applies to all bases? By the way, the above totally-cylindrical spacetime has a natural expression in terms of "octonion space", i.e., the Cayley algebra whose elements are two ordered quaterions x1 = i R1 cos x/R1 x2 = i R1 sin x/R1 x3 = j R2 cos y/R2 x4 = j R2 sin y/R2 x5 = k R3 cos z/R3 x6 = k R3 sin z/R3 x7 = R4 cos t/R4 x8 = R4 sin t/R4 Thus each point (x,y,z,t) in 4D spacetime represents two quaterions q1 = x1 + x3 + x5 + x7 q2 = x2 + x4 + x6 + x8 To determine the absolute distances in this 8D space we again consider the eight coordinate differentials, exemplified by d x1 = i R1 (-sin(x/R1)) (1/R1) (dx) (using the rule for total differentials) so the squared differentials are exemplified by (d x1)^2 = - sin^2(x/R1) (dx)^2 Adding up the eight squared differentials to give the square of the absolute differential interval leads again to the locally Lorentzian 4D metric (ds)^2 = (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2
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